A person is riding a bicycle, and its wheels have an angular velocity of 26.0 rad/s. Then, the brakes are applied and the bike is brought to a uniform stop. During braking, the angular displacement of each wheel is 17.6 revolutions. (a) How much time does it take for the bike to come to rest? (b) What is the anguar acceleration (in rad/s2) of each wheel?

Question

A person is riding a bicycle, and its wheels have an angular velocity of 26.0 rad/s. Then, the brakes are applied and the bike is brought to a uniform stop. During braking, the angular displacement of each wheel is 17.6 revolutions. (a) How much time does it take for the bike to come to rest? (b) What is the anguar acceleration (in rad/s2) of each wheel?

Do I need to convert revolutions to radians first?

Physics(Please respond) - Elena, Tuesday, June 5, 2012 at 11:31am
Equations for decelerated motion are
ù=ùₒ - å•t,
ö = ùₒ•t - å•t²/2.

Since ù =0, ö = 2•ð•N, where N =17.6 rev.
these equatuions are
0=ùₒ-å•t, …………………… (1)
2•ð•N = ùₒ•t - å•t²/2 …………(2)
From (1)
å = ùₒ/t……………………… (3)
Plug it in (2) and obtain
t= 4•ð•N/ ùₒ.
Then determine angular acceleration using (3).

so first I have to do equation 2? So
2*pi*17.6 = I am not sure what wo is. Would it be the 26.0?

Answer 1

ε = ωₒ/t

(3) ->(2)
2•π•N = ωₒ•t - ε•t²/2 = ωₒ• t – (ωₒ/t) •t²/2 = ωₒ• t/2,
t= 4•π•N/ ωₒ =4•π•17.6/26 =8.5 s
ε = ωₒ/t = 26/8.5=3.06 rad/s²

Answer 2

Yes, in order to determine the time it takes for the bike to come to rest, you need to use Equation (2). Let's go through the equation step by step.

The equation is:

2πN = ùₒt - åt²/2

Here, ùₒ represents the initial angular velocity of the wheels, and N represents the angular displacement in revolutions. In this case, N is given as 17.6 revolutions.

To solve for ùₒ, the initial angular velocity, you need to know the value. In the question, it is given that the initial angular velocity is 26.0 rad/s. So ùₒ = 26.0 rad/s.

Now, substitute the values into the equation:

2π * 17.6 = 26.0t - åt²/2

Simplifying:

(2π * 17.6) = 26.0t - (å/2)t²

(2π * 17.6) = (26.0 - (å/2))t²

Now, to solve for t, we need to isolate it on one side of the equation. So let's rearrange the equation:

(t²)((å/2) - 26.0) = (2π * 17.6)

t² = (2π * 17.6) / ((å/2) - 26.0)

Finally, solve for t:

t = √((2π * 17.6) / ((å/2) - 26.0))

Once you find the value of t, you can move on to determining the angular acceleration (å), using Equation (3).

Remember that å = ùₒ/t, where ùₒ is the initial angular velocity and t is the time it takes for the bike to come to rest. Substitute the known values into the equation to find the angular acceleration.